It depends on how cold it is. If the temps are hovering in the low teens are lower, I have found that all temps will start to stick. I've even had problems with cold temp waxes as most of them are designed for temps above 0. Swix and a few others make very cold weather waxes which work great in those super cold temps.
Anyway, back to your questions. It never hurts to clean your base before a waxing. It's not like you have to rub it out (heh) big time. Just get the base clean on it and wipe it down. Then go ahead and wax the board. If you have any wax residue, say from where you didn't scrape it off the first time, by all means scrape it off. Otherwise, spray on the base cleaner and wipe it off. You'll get ride of dust, particles, and such by doing that before you wax. If not, you're fine to just wax it. I got a hunch your board will perform a lot better with the cold weather wax on.